A continuous model for investigation of physically nonlinear deformation of orthotropic sandwich plates

A phenomenological yield condition for quasi-brittle and plastic orthotropic materials with initial stresses is suggested. All components of the yield tensor are determined from experiments on uniaxial loading. The reliability estimates of the criterion suggested is discussed. For a plastic material without initial stresses, the given condition transforms into the Marin—Hu criterion. The defining equations of the deformation theory of plasticity with isotropic and “anisotropic” hardening, associated with the yield condition suggested, are obtained. These equations are used as the basis for a highly accurate nonclassical continuous model for nonlinear deformation of thick sandwich plates. The approximations with respect to the transverse coordinate take into account the flexural and nonflexural deformations in transverse shear and compression. The high-order approximations allow us to model the occurrence of layer delamination cracks by introducing thin nonrigid interlayers without violating the continuity concept of the theory. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. pp. 329–340, May–June, 2000.     

简单剪切振荡现象及弹塑性本构的限制条件

基于在真应力空间刻划弹塑性物质的强化、软化和理想塑性特性,本文以刚塑性随动强化模型为例说明产生简单剪切振荡现象,是与模型在简单剪切变形下,其强化和软化特性发生交替变化的现象有关。为使弹塑性本构模型更符合实际,要求它们必须满足如下条件:即对任意弹塑性加载变形过程,本构模型所给出的应力应该是非振荡的,所描述的强化和软化特性,不存在从软化阶段至强化阶段的过渡。     

Mathematical modeling of the stress state of a transverse plastic layer in a round rod

We construct mathematical models of plastic deformations of a continuous round rod containing a transverse (less strong) inhomogeneous layer under an axial load. We thoroughly study the local strengthening of such layers by involving the base material of the rod in the plastic deformation process. We obtain explicit formulas for the critical stress states in the layer and the critical axial load on the rod.     

正交各向异性材料的混合硬化弹塑性本构方程

建立了混合硬化正交各向异性材料的屈服准则,进而推导了与之相关的塑性流动法则.根据简单应力状态的实验曲线,可得到广义等效应力-应变关系.初始屈服曲面与材料的弹性常数有关,材料退化为各向同性且只考虑各向同性硬化时,屈服函数退化为Huber-Mises屈服函数,相关的本构方程退化为Prandtl-Reuss方程.     

Kinematics of finite elastoplastic deformations

In this paper we review various approaches to the decomposition of total strains into elastic and nonelastic (plastic) components in the multiplicative representation of the deformation gradient tensor. We briefly describe the kinematics of finite deformations and arbitrary plastic flows. We show that products of principal values of distortion tensors for elastic and plastic deformations define principal values of the distortion tensor for total deformations. We describe two groups of methods for decomposing deformations and their rates into elastic and nonelastic components. The methods of the first group additively decompose specially built tensors defined in a common basis (initial, current, or “intermediate”). The second group implies a certain relation connecting tensors that describe elastic and plastic deformations. We adduce an example of constructing constitutive relations for elastoplastic continuums at large deformations from thermodynamic equations.     

The limit equilibrium of an anisotropic medium under the general plasticity condition

A theory of the limit equilibrium of an anisotropic medium under the general plasticity condition in the plane strain state is developed. The proposed yield criterion (the limit equilibrium condition) is obtained by combining the von Mises–Hill yield criterion of an ideally plastic anisotropic material and Prandtl's limit equilibrium condition for a medium under the general plasticity law. It is shown that the problem is statically determinate, i.e., if the boundary conditions are specified in stresses, the stress state in plastic region can only be obtained using equilibrium equations. It is established that the equations describing the stress state are hyperbolic and have two families of characteristic curves that intersect at variable angles. In deriving the equations describing the velocity field, the material is assumed to be rigid plastic, and the associated law of flow is applied. It is shown that the equations for the velocities are also hyperbolic, and their characteristic curves are identical with those of the equations for stresses. However, the directions of the principal values of the stress and strain rate tensors are different due to the anisotropy of the material. The characteristic directions differ from the isotropic case in that the normal and tangential components of the stress tensor do not satisfy the limit conditions. It is established that the equations obtained allow of partial solutions, and in this case, at least one family of characteristic curves consists of straight lines. The conditions along the lines of discontinuity of the velocity are investigated, and it is shown that, as in the isotropic case, these are characteristic curves of the system of governing equations. In the anisotropic formulation, the well-known Rankine problem of the limit state of a ponderable layer is solved. From an analysis of the velocity field it is shown that plastic flow of the entire layer is possible only for a slope angle equal to the angle of internal friction. For slope angles less than the angle of internal friction, the solutions obtained are solutions of problems of the pressure of the medium on the retaining walls. The change in this pressure as a function of the parameters of anisotropy is investigated, and turns out to be significant.     

环壳理论与直交异性板理论在计算三圆弧波纹膜片上的比较

有人用直交异性板的理论计算[1]波纹板(膜片)的弹性位移,所得结果曾与实验对照比较满意.但是并没有人认真分析波纹数目、形状对直交异性板的弹性位移和应力分布的数量上的影响,以致未能对用直交异性板理论计算波纹板的范围作明确的说明.以前只是较一般地说,直交异性板理论用于计算波纹数目较多的膜片弹性特性(弹性位移与外力的关系)较满意,计算应力误差较大.本文利用环壳理论[2][3]分析了对称和不对称三圆弧波纹膜片的位移与应力,并与直交异性理论所得结果比较,明确了直交异性板理论的应用范围.     

Description of plastic deformation of structural materials in triaxial loading

A model of nonassociated plasticity is put forward for initially isotropic materials deforming with residual changes in volume under the action of triaxial normal stresses. The model is based on novel plastic loading and plastic potential functions, which define closed, convex, every where smooth surfaces in the 6D space of symmetric second-rank stress tensors. By way of example, the plastic deformation of a cylindrical concrete specimen wrapped with a CFRP tape and loaded in axial compression is described. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 183–194, March–April, 2008.     

Results of an experimental investigation of the residual stresses in wound glass-reinforced plastics

The results of an experimental investigation of the residual stresses in wound glass-reinforced plastic rings are presented. The residual stresses were determined by the Davidenkov method. The dependence of the maximum tensile and compressive circumferential stresses on ring thickness and polymerization temperature is investigated. The experimental data are compared with the results of calculations based on the theory proposed in [1, 2].Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1116–1119, November–December, 1970.     

Some mechanical properties of film-reinforced plastics

Model cylindrical shells have been wound from glass and Dacron (Mylar) film strip in accordance with a technique developed by the authors. A layer of glass-reinforced plastic, oriented in the direction of the circumferential tensile stresses, was applied to the outer cylindrical surface of the shells. Certain mechanical properties of the film plastics and their ability to cooperate with the glass-reinforced plastic outer layer were investigated by subjecting the shells to internal liquid pressure. The possibility of obtaining impermeable shells, equally strong in the axial and circumferential directions, was confirmed.Bauman Moscow Higher Technical College. Translated from Mekhanika Polimerov, No. 1, pp. 135–139, January–February, 1971.     

Experimental testing of the applicability of a fourth degree polynomial in describing surfaces of equicritical planar stress distributions in glass-reinforced plastics

The possibility of using a fourth degree polynomial for calculating the anisotropy of glass-reinforced plastics in the planar stress state is evaluated experimentally. The experiments were carried out on industrial specimens of glass-reinforced plastic tubes loaded with an internal pressure, and subjected to an axial tensile force and a torque. Surfaces of equicritical planar stress distribution were constructed for the material of the glass-reinforced plastic tubes. The experimental and theoretical values of the breaking stresses were compared statistically. Most of the experimental values of the breaking stresses were found to lie within the confidence intervals of their theoretical values.S. M. Kirov Timber Technology Academy, Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 284–294, March–April, 1970.     

Complete elastic–plastic stress asymptotic solutions near general plane V-notch tips

An efficient method is developed to determine the multiple term eigen-solutions of the elastic–plastic stress fields at the plane V-notch tip in power-law hardening materials. By introducing the asymptotic expansions of stress and displacement fields around the V-notch tip into the fundamental equations of elastic–plastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions are established. Then the interpolating matrix method is employed to solve the resulting nonlinear and linear ODEs. Consequently, the first four and even more terms of the stress exponents and the associated eigen-solutions are obtained. The present method has the advantages of greater versatility and high accuracy, which is capable of dealing with the V-notches with arbitrary opening angle under plane strain and plane stress. In the present analysis, both the elastic and the plastic deformations are considered, thus the complete elastic and plastic stress asymptotic solutions are evaluated. Numerical examples are shown to demonstrate the accuracy and effectiveness of the present method.     

Modeling residual stresses in arterial walls based on anisotropic growth

With the aim of obtaining a general local formulation for anisotropic growth in soft biological tissues, a model based on the multiplicative decomposition of the growth tensor is formulated. The two parts of the growth tensor are associated with the main anisotropy directions. Together with an anisotropic driving force, the model enables an effective stress reduction by including growth-induced residual stresses, which is demonstrated in a numerical example of an idealized arterial segment. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The numerical analysis of the trapezoidal thread rolling process

The thread rolling is difficult technological process. Improve quality and contemporary reduce manufacture cost of the trapezoidal thread requires acquaintances of physical phenomena observed in the contact zone between rolls and deform work-pieces. Therefore, in this paper the physical and mathematical models of deformations (displacements and strains) and stress in the cold process of trapezoidal thread rolling, were developed. The process is considered as a geometrical and physical non-linear, initial as well as boundary value problem. The phenomena on a typical incremental step were described using a step-by-step incremental procedure, with an updated Lagrangian formulation. The state of strains was described by Green-Lagrange's tensor, while the state of stress by the second symmetrical Pioli-Kirchhoff's tensor. The object was treated as an elastic (in the reversible zone) and visco-plastic body (in non-reversible zone) with mixed hardening. The variational equation of motion in three dimensions for this case was proposed. Then, the finite elements methods (FEM) and dynamic explicit method (DEM) were used to obtain the solution. The application developed for in the ANSYS programme, which provides a complex time analysis for displacement, strains and stresses occurring in the object. The recommendations concern modeling the trapezoidal thread rolling process, where reduce degrees of freedom in numerical model is very important and provide convergence calculated results for maximum stress and strain values in the thread surface layer, were elaborated. The influence a various process conditions on the states deformation and stress for examples calculations, were presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deformation Behavior Modeling of SMAs Under Cyclic Loading Based on Rational Interpolation北大核心CSCD

提出了一个有限弹塑性模型,用来模拟形状记忆合金(shape memory alloys,SMAs)在循环荷载下的变形行为.首先,通过分析上下屈服阶段形函数的特点,利用有理插值方法给出循环荷载下的应力-应变形函数显式表达,可以精确匹配任意形状的实验数据;其次,基于对数客观率,构建了有限弹塑性J2流模型,耦合了屈服中心的移动和屈服面的增大;再次,从单轴情况出发,推导得到了单个循环下的三个硬化函数显式表达,再引入局部因子和多轴扩展不变量,构造了光滑统一且多轴有效的硬化函数;最后,将模型得到的结果与经典实验结果比较,证明了新方法的有效性.该文创新点如下:第一,通过改进传统的背应力演化方程,使得新模型产生强烈的Bauschinger效应,从而使新方法具备模拟SMAs特殊变形行为的能力;第二,新的光滑统一硬化函数在单个循环下会自动退化,得到精确符合实验数据的结果;第三,利用本构方程推导得到有效塑性功演化规律,而有理插值得到的形函数中包含依赖有效塑性功的参数,给出这些参数方程以后使得模型具备了预测变形的能力.     

Computation of thermal residual stresses in optic fibers

We propose an analytic method of determining all the components of the residual stress tensor in optic fibers. The method is based on solving a plane problem for a cylindrical structure with singular stresses. We obtain closed-form solutions of the problem in the case of a piecewise-constant distribution of free deformations that model the presence of inclusions in the fiber with different values of the thermal coefficient of expansion. We also consider inclusions with cross sections in the shape of a circle, a central ellipse and a central annular sector. We describe the results obtained on this basis in the computation of residual thermal stresses.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 79–83.     

Influence of heating on the stability of fiberglass reinforced plastic shells during axial compression

Results are described of the experimental investigation of the patterns of change of carrying capacity and deformation of smooth thin-walled round cylindrical shells made from fiberglass reinforced plastic based on phenol-formaldehyde binder, loaded by an axial compressing force in a temperature field uniform with respect to the thickness of the wall and the surface. The experimental data are compared with calculated ones obtained according to the formulae of the theory of orthotropic shells. The influence of circumferential compressing stresses arising in the junction zone of the heated wall with less heated thicker parts at the edges (ribs) on the carrying capacity of the shells is evaluated.Professor N. E. Zhukovskii Central Aerodynamics Institute, Moscow Oblast. Translated from Mekhanika Polimerov, No. 5, pp. 838–845, September–October, 1972.     

Wrinkle-free solutions in the theory of annular elastic membranes

Rotationally symmetric deformations of a flat annular elastic membrane under a gravitational force are studied, with prescribed radial stresses or horizontal displacements at the edges. The small-finitedeflection theory of Föppl-Hencky as well as a simplified version of Reissner's static first approximation theory of thin shells of revolution are applied which lead to consider a single, second-order, ordinary differential equation for the derivation of the principal stresses in the membrane. Using analytical methods, the range of those boundary data is determined for which the solutions of the differential equation are wrinkle free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.